FOM: Proper Names and the Diagonal Proof
William Tait
wwtx at uchicago.edu
Wed Jun 26 15:53:57 EDT 2002
Dean Buckner (6/24) wrote
>Cantor's proof has it otherwise. It says that we cannot "match" a set of
>ordinary names (numerals) to a set of names formed by infinite decimal
>expansion. Each row is "complete": it is a proper name formed out of an
>"infinite" collection of digits. It assumes not just an infinite collection
>of names, but an infinite collection of names with infinite number of parts.
>But it's not clear which object is named by any name in the the sequence,
>since it is not clear what the name actually is. Even if we suppose that
>"the square root of two" names something, what does "1.23245467 ..."
>name,when the dots signify not a specific method of carrying on the
>sequence, but no specific method at all?
>
>I'm happy to assume that, given that a meaningful expression of English can
>formed using a finite sequence of expressions, using words drawn from a
>finite vocabulary, we can signify a different thing with each different
>expression. I'm not happy to assume any more than that, unless I
>know what I'm meant to assume.
>
>I'm not arguing against Cantor's proof, I'm saying I don't understand what
>it's meant to prove.
Cantor's theorem states that, given any enumeration of real numbers
in an interval (a,b), there exists a real number in that interval
which is not in the enumeration. (The proof, in 1873, incidentally,
was not by the diagonal argument. The latter was used in 1890 to
prove a more general theorem.) There is no reference in the statement
or in Cantor's proof (a nested sequence of intervals argument) to
names or expressions. The same is true of his statement that the set
of two-valued functions defined on a set M has power greater than
that of M and of his diagonal argument by means of which he proved
it. In particular, nothing hangs on whether or not one is willing to
call an infinite decimal expansion of an irrational number a `name'
of it.
I think that Dean Buckner has reformulated Cantor's statement and
argument and then attacked the terms of his own formulation.
I suppose that another way to look at it is that Dean is trying to
understand Cantor's theorem while avoiding the `set-theoretic'
language in which it formulated and proved. It took around 24 hundred
years (if we start only with the classical Greeks) for the notion of
set to become clear---and by `clear', I mean to have an established
use, which we may express by a set of axioms (to be sure,
open-ended). Why should one want to relive that history?
Respectfully,
Puzzled in Chicago (alias: Bill Tait)
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